Multipurpose Reservoir Operation Using Particle
Swarm Optimization
D. Nagesh Kumar1 and M. Janga Reddy2
Abstract: This paper presents an efficient and reliable swarm intelligence-based approach, namely elitist-mutated particle swarm optimization 共EMPSO兲 technique, to derive reservoir operation policies for multipurpose reservoir systems. Particle swarm optimizers are
inherently distributed algorithms, in which the solution for a problem emerges from the interactions between many simple individuals
called particles. In this study the standard particle swarm optimization 共PSO兲 algorithm is further improved by incorporating a
new strategic mechanism called elitist-mutation to improve its performance. The proposed approach is first tested on a hypothetical
multireservoir system, used by earlier researchers. EMPSO showed promising results, when compared with other techniques. To show
practical utility, EMPSO is then applied to a realistic case study, the Bhadra reservoir system in India, which serves multiple purposes,
namely irrigation and hydropower generation. To handle multiple objectives of the problem, a weighted approach is adopted. The results
obtained demonstrate that EMPSO is consistently performing better than the standard PSO and genetic algorithm techniques. It is seen
that EMPSO is yielding better quality solutions with less number of function evaluations.
DOI: 10.1061/共ASCE兲0733-9496共2007兲133:3共192兲
CE Database subject headings: Reservoir operation; Optimization; Irrigation; Hydroelectric power generation.
Introduction
In reservoir operation problems, to achieve the best possible performance of the system, decisions need to be taken on releases
and storages over a period of time considering the variations in
inflows and demands. In the past, various researchers applied different kinds of mathematical programming techniques like linear
programming, dynamic programming, nonlinear programming
共NLP兲, etc., to solve such reservoir operation problems. An extensive review of these techniques can be found in Loucks et al.
共1981兲, Yakowitz 共1982兲, Yeh 共1985兲, and Wurbs 共1993兲. But as
far as reservoir operation is concerned, no standard algorithm is
available, as each problem has its own individual physical and
operational characteristics 共Yeh 1985兲.
In the case of multipurpose reservoir operation, the goals are
more complex than for single purpose reservoir operation and
often involve various problems such as insufficient inflows and
larger demands. In order to achieve the best possible performance
of such a reservoir system, a model should be formulated as close
to reality as possible. In this process, the model is expected to
solve problems having nonlinearities and nonconvexities in their
domain. For example, a typical hydropower production function
is complex, with nonlinear relationships in objectives and con1
Associate Professor, Dept. of Civil Engineering, Indian Institute of
Science, Bangalore 560 012, India. E-mail: nagesh@civil.iisc.ernet.in
2
Research Scholar, Dept. of Civil Engineering, Indian Institute of
Science, Bangalore 560 012, India. E-mail: mjreddy@civil.iisc.ernet.in
Note. Discussion open until October 1, 2007. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on January 19, 2005; approved on April 3, 2006. This
paper is part of the Journal of Water Resources Planning and Management, Vol. 133, No. 3, May 1, 2007. ©ASCE, ISSN 0733-9496/2007/3192–201/$25.00.
straints. So the linear programming methods cannot be used. The
dynamic programming approach faces the additional problem of
the curse of dimensionality, whereas the nonlinear programming
methods have the limitations of slow rate of convergence, requiring large amounts of computational storage and time compared
with other methods 共Yeh 1985兲. Also, often NLP results in local
optimal solutions.
In spite of development of many conventional techniques for
optimization, each of these techniques has its own limitations. To
overcome those limitations, recently metaheuristic techniques are
being used for optimization. By using these techniques, the given
problem can be represented more realistically. These also provide
ease in handling the nonlinear and nonconvex relationships of the
formulated model. Genetic algorithms 共GAs兲 共Goldberg 1989兲
and particle swarm optimization 共PSO兲 共Eberhart and Kennedy
1995兲 are some of the techniques in this category. These evolutionary algorithms search from a population of points, so there is
a greater possibility to cover the whole search space and reaching
the global optimum.
GA is one of the population-based search techniques, which
works on the concept of “survival of the fittest” 共Goldberg 1989兲.
In the field of water resources, in earlier studies, few applications
of the GA technique to derive reservoir operating policies have
been reported 共Oliveira and Loucks 1997; Chang and Chen 1998;
Wardlaw and Sharif 1999兲 and they illustrated the utility of evolutionary techniques for reservoir operation problems. Though
GA has many advantages over conventional methods, it also has
some drawbacks, such as slow rate of convergence and requiring
a large number of simulations to arrive at an optimum solution.
Recently researchers reported that PSO is showing some improvement on these issues, which use the local and global search
capabilities of the algorithm, to find better quality solutions with
less computational time 共Salman et al. 2002兲.
To improve the performance of the standard PSO, in this study,
a new strategic mechanism called elitist mutation is incorporated
192 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007
in to the algorithm. Using this mechanism, elitist-mutated particle
swarm optimization 共EMPSO兲 is propounded to derive operational policies for multipurpose reservoir systems. To test the
usefulness of EMPSO for obtaining reservoir operating polices,
first it is tested for a hypothetical reservoir system, which was
used by earlier researchers. To show its practical utility, EMPSO
is then applied to an existing reservoir system, namely the Bhadra
reservoir system in Karnataka State, India. To compare the performance of EMPSO, the previous two case studies are also
solved using standard PSO and GA techniques and the results are
discussed.
Particle Swarm Optimization
The term “swarm intelligence” is used to describe algorithms and
distributed problem solvers, which was inspired by the collective
behavior of insect colonies and other animal societies. Under this
prism, PSO is a swarm intelligence method for solving optimization problems. Particle swarm optimization, originally proposed
by Eberhart and Kennedy 共1995兲, is a population-based heuristic
search technique, inspired by social behavior of bird flocking.
PSO shares many similarities with evolutionary computation
techniques such as GA. PSOs are initialized with a population of
random solutions and search for optima by updating generations.
However, in contrast to methods like GAs, in basic PSO, no operators inspired by natural evolution are applied to extract a new
generation of candidate solutions. Instead, PSO relies on the exchange of information between individuals 共particles兲 of the
population 共swarm兲. In effect, each particle adjusts its trajectory
towards its own previous best position and towards the current
best position attained by any other member in its neighborhood
共Parsopoulos and Vrahatis 2002兲. By using the PSO technique,
it will be easy to handle nonlinearity and nonconvexity of
the problem domain; the search does not depend on the initial
population; and overcomes problems of local optima that are
common in some conventional nonlinear optimization techniques
共Salman et al. 2002兲. The PSO technique has successful demonstration for numerical optimization and reported that PSO
is an attractive alternative for global optimization problems
共Parsopoulos and Vrahatis 2002兲. It has also been found that PSO
is outperforming other heuristic search methods such as GAs
共Abido 2002; Salman et al. 2002兲.
PSO Algorithm
If the search space is D-dimensional, the ith individual 共particle兲
of the population 共swarm兲 can be represented by a D-dimensional
vector, Xi = 共xi1 , xi2 , . . . , xiD兲T. The velocity 共position change兲 of
this particle can be represented by another D-dimensional vector
Vi = 共vi1 , vi2 , . . . , viD兲T. The best previously visited position of the
ith particle is denoted as Pi = 共pi1 , pi2 , . . . , piD兲T. Defining g as the
index of the best particle in the swarm 共i.e., the gth particle is the
best兲, and superscripts denoting the iteration number, the swarm
is manipulated using the following two equations:
n
n
n
n n
n n
vn+1
id = ␹关␻vid + c1r1共pid − xid兲 + c2r2共pgd − xid兲兴
xn+1
id
=
xnid
+
vn+1
id
共1兲
共2兲
where d⫽1,2,. . . , D; i⫽1,2,. . . , N; N⫽size of the swarm;
␹⫽constriction coefficient; ␻⫽inertial weight; c1 and c2⫽positive constant parameters called acceleration coefficients;
Fig. 1. Pseudocode of elitist-mutated particle swarm optimization
algorithm
r1,r2⫽random numbers, uniformly distributed in 关0,1兴; and
n⫽iteration number.
The general effect of Eq. 共1兲 is that each particle oscillates in
the search space, between its previous best position and the best
position of its best neighbor, attempting to find the new best point
in its trajectory. Proper fine-tuning of the parameters c1 and c2, in
Eq. 共1兲, may result in faster convergence of the algorithm, and
alleviation of the problem of local minima. The role of inertial
weight ␻ is to control the impact of the previous velocities on the
current one. A large inertial weight facilitates global exploration
共searching new areas兲, while a small weight tends to facilitate
local exploration. Hence selection of a suitable value for the inertial weight ␻ usually helps in reduction of the number of iterations required to locate the optimum solution 共Parsopoulos and
Vrahatis 2002兲.
In basic PSO algorithms, the constriction factor ␹ is taken as 1.
In later developments of the PSO technique, it is observed that if
the particle’s velocity is allowed to change without bounds, the
swarm will never converge to an optimum, since subsequent oscillations of the particle will be larger. To control the changes in
velocity, the constriction factor ␹ was introduced into the PSO
algorithm in Clerc 共1999兲, to force the swarm to converge better.
By using Eqs. 共1兲 and 共2兲 the velocities and particle positions are
updated repeatedly over the iterations, to get at the optimal solution. Even though PSO is faster in finding quality solutions, compared to other evolutionary computation techniques, it faces some
difficulty in obtaining better quality solutions while exploring
complex functions. It may face premature convergence and suffer
from poor fine-tuning capability of the final solution.
Elitist-Mutated Particle Swarm Optimization
To overcome the problems explained earlier and to improve the
performance of the PSO technique, an elitist-mutation strategy is
introduced into the algorithm in this study. This elitist process
replaces the worst particle solutions by the best solution among
the swarm, after performing mutation mechanism on the best particle. This process of random perturbation tries to improve the
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007 / 193
solution, by maintaining diversity in the population, and explores
the new regions in the whole search space. This strategy replaces
the position vectors of a predefined number of least ranked particles in the swarm with mutated positions of the global best
particle in each iteration. Pseudocode of the EMPSO algorithm is
presented in Fig. 1, describing the main steps involved. In the
EMPSO methodology, the elitist-mutation step is computed as
follows. First the fitness function of particles is sorted in ascending order and the index number for the respective particles is
obtained, then elitist-mutation is performed on worst fitness
particles in the swarm. Let NMmax⫽number of particles to be
elitist-mutated; g⫽index of global best particle; pem⫽probability
of mutation; ASF⫽index of sorted population; rand⫽uniformly
distributed random number U共0 , 1兲; randn⫽Gaussian random
number N共0 , 1兲; and VR关d兴⫽range of decision variable d
Fig. 2. Four-reservoir system considered for Case Study I 关adapted
from Larson 共1968兲兴
For i⫽1 to NMmax
l = ASF关i兴
For d = 1 to dim
if 共rand⬍ pem兲
X关l兴关d兴 = P关g兴关d兴+0.1* VR关d兴*randn
else
X关l兴关d兴 = P关g兴关d兴
If the mutated value exceeds the bounds, then it is limited to
the upper or lower bound. During this elitist-mutation step, the
velocity vector of the particle is unchanged. The step-by-step
procedure of EMPSO technique for the reservoir operation
problem is explained in the following.
1. Initialization of position and velocity vectors of the swarm:
For a reservoir operation problem, the search space will be
equal to the total number of release combinations. Each decision variable represents a parameter to be optimized in the
model. The initial positions of all particles have to be generated randomly within the limits specified for each decision
variable. For the reservoir operation problem, the initial
searching points are the initial random release values. Each
particle is initialized with random position vectors, Xi共0兲, and
random velocity vectors, Vi共0兲, for all i⫽1,2,…,N.
2. Initial evaluation of fitness function: The fitness of each particle is evaluated using the objective function of the problem.
Then the best value out of all the fitness values is searched.
Two “best” values are defined, the global best and the personal best. The global best Pg is the highest fitness value
among all the particles in an entire run 共best solution so far兲,
and the personal best Pi is the highest fitness value of a
specific particle up to the current iteration.
3. Modification of each searching point: Using the global best
and the local best of each particle up to the current iteration,
the searching point of each particle is modified using the
change in velocity as given by Eq. 共1兲 and the positions of
the particles are updated using Eq. 共2兲.
4. Evaluation of fitness function: After modification of the particle positions, the fitness of each particle is evaluated using
the objective function of the problem.
5. Perform elitist mutation: The fitness values are ranked in the
order of their fitness values. For the selected number of least
ranked particles 共which is equal to the number of particles to
perform elitist mutation in iteration兲, the respective particle
position vectors are replaced with the new position vectors
obtained after performing variable-wise mutation on the best
particle position vector.
6.
Updating the global and the local bests: The Pg and Pi values have to be updated according to the new fitness values. If
the best fitness value of a particular particle in the swarm is
better than the current Pg, then Pg is to be changed to the
value of the searching point of the corresponding particle
contributing to this best fitness value. Similarly the local best
of other particles in the population should be changed accordingly if the present fitness function value is better than
the previous.
7. Termination criteria: Repeat steps 共3兲 to 共6兲 until either the
pre-set maximum number of iterations is reached or no significant improvement is observed over a prespecified number
of iterations.
To compare the performance of EMPSO, the real coded
GA developed by Kanpur Genetic Algorithms Laboratory, IIT
Kanpur, India 共KanGAL, http://www.iitk.ac.in/kangal/soft.htm兲 is
used in this study. The performance of GA is tested for benchmark problems and is performing quite well as reported in the
literature 共Deb 2000兲. However, source code of the GA program
is modified to suit the present problem and then used to run the
model with the same formulation as that used for EMPSO.
The next section presents application of the EMPSO technique
for two case studies, one hypothetical and one real life reservoir
system.
Case Study I
In this study, a hypothetical four-reservoir system 共Larson 1968兲
is used to test the efficiency of the EMPSO technique, to derive
operating policies for the multireservoir system. The fourreservoir system considered for this case is shown in Fig. 2. For
this hypothetical reservoir system in the past, Larson 共1968兲 applied dynamic programming with successive approximation
共DPSA兲 algorithm, Heidari et al. 共1971兲 applied discrete differential dynamic programming 共DDDP兲, and Kumar and Singh 共2003兲
applied folded dynamic programming 共FDP兲 to evaluate the performance of their respective proposed techniques. Presently, the
EMPSO technique is applied to the same hypothetical system
with the same objective function, constraints, and related data as
given in Larson 共1968兲. The same reservoir system is also solved
using standard PSO and GA techniques to facilitate comparison.
For the EMPSO model, the parameters chosen after thorough
sensitivity analyses are constriction coefficient 共␹兲 = 0.9; inertia
weight ␻ = 1.0; PSO constant parameters c1 = 1.0, c2 = 0.5;
194 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007
Table 1. Performance of Six Models for the Hypothetical Four-Reservoir
System
Model
Source
Best fitness
value
Number
of function
evaluations taken
DPSA
Larson 共1968兲
401.3
—
DDDP
Heidari et al. 共1971兲
401.3
—
FDP
Kumar and Singh 共2003兲
399.06
—
Presently applied
399.7
748,000
PSOa
Presently applied
401.3
2,279,500
GAa
Presently applied
401.3
325,400
EMPSOa
a
The best solution obtained for ten trial runs with the respective
algorithm.
probability of elitist mutation 共Pem兲 = 0.2; size of the swarm
N = 2,000; maximum number of iterations⫽500; and the size of
elitist-mutated particles⫽38. For this hypothetical reservoir system, elitist mutation is performed for all the iterations. To run the
standard PSO model, the same parameters are used except the
elitist-mutation step. For the GA model, the parameters used are
as follows: crossover probability 共Pc兲 = 0.9 and a variable-wise
mutation probability 共Pm兲 = 0.2; distribution index for simulated
binary crossover⫽10 and that for mutation operator⫽100; size of
the population⫽500; and number of generations⫽5,000.
To check the performance of the proposed technique, the models are run for ten random trials. The performance of six models
for the hypothetical four-reservoir system is shown in Table 1,
which compares the best fitness value obtained by the earlier
techniques and also with presently applied models with respect to
the number of function evaluations taken for that solution. The
proposed EMPSO technique yielded best solution as 401.3, which
is also equal to the global optimal solution reported in the literature. For ten trial runs, the mean fitness values obtained and number of function evaluations taken are 401.18 共447,830兲, 396.16
共683,400兲, and 400.84 共2,408,750兲, respectively, for EMPSO,
standard PSO, and GA. These results clearly show that the
EMPSO technique is giving the best fitness value, with least number of function evaluations as compared to other models.
Fig. 3. Schematic diagram of the Bhadra reservoir system
loamy soil, except in some portions of the right bank canal area,
which also consists of black cotton soil. The major crops grown in
the command area are paddy, sugarcane, permanent garden, and
semidry crops.
The data of monthly inflows and other details were collected
from the Water Resources Development Organization, Bangalore
covering a period of 69 years 共from 1930–1931 to 1998–1999兲.
The monthly crop water requirements were calculated using the
FAO Penman–Monteith method. In addition to irrigation and hydropower, a mandatory release of 9 ⫻ 106 m3 / month to downstream is to be made in each month to meet the water quality
requirements.
Mathematical Model Formulation
The two objectives considered in the model are minimization of
irrigation deficits and maximization of hydropower generation.
These two are conflicting objectives, as one tries to minimize the
irrigation deficits, requiring more water to be released to satisfy
irrigation demands and the other tries to maximize hydropower
production, which requires higher level of storage to be maintained in the reservoir to produce more power. The two competing
objectives of the system are expressed as,
Minimize sum of squared deficits for irrigation annually
12
Minimize SQDV =
Case Study II
To evaluate the practical utility of the proposed technique, an
existing reservoir system, namely the Bhadra reservoir system, is
taken up as a case study for developing optimal reservoir operation policies. The Bhadra Dam is located at latitude 13° 42⬘ N and
longitude 75° 38⬘20⬙ E in Chikmagalur District of Karnataka
State, India. The average annual rainfall in the basin is 2,320 mm
with 90% rainfall occurring during the monsoon period 共June–
November兲. The Bhadra reservoir is a multipurpose reservoir for
irrigation and hydropower generation, in addition to mandatory
releases, to maintain water quality downstream.
The reservoir has an active storage capacity of 1,784
⫻ 106 m3 and the water-spread area at full reservoir level is
117 km2. The reservoir provides water for irrigation of 6,367 and
87,512 ha under left and right bank canals, respectively. Also
under this project there are three sets of turbines, one set each on
the left bank canal and the right bank canal and the other set at the
river bed level of the dam, for generating hydropower. Fig. 3
shows the schematic diagram of the Bhadra reservoir system. The
irrigated area is spread over the districts of Chitradurga, Shimoga,
Chikmagalur, and Bellary and comprises predominantly of red
兺
t=1
12
共Dl,t − Ql,t兲2 +
共Dr,t − Qr,t兲2
兺
t=1
共3兲
where SQDV⫽squared deviations of irrigation demand and releases; Dl,t and Dr,t⫽irrigation demands for the left bank canal
and right bank canal command areas, respectively, in period t
共106 m3兲; and Ql,t and Qr,t⫽releases made into the left and right
bank canals, respectively, in period t 共106 m3兲.
Maximize annual energy production
12
Maximize E =
p共Ql,tHl,t + Qr,tHr,t + Qb,tHb,t兲
兺
t=1
共4兲
where E⫽energy produced 共⫻106 kW h兲; p⫽power production
coefficient; Qb,t⫽release made to riverbed turbine in period t
共106m3兲; and Hl,t, Hr,t, Hb,t⫽net heads available 共m兲 to left bank,
right bank, and bed turbines, respectively, in period t.
Subject to the following constraints:
• Mass balance equation
St+1 = St + It − 共Ql,t + Qr,t + Qb,t + EVPt + OVFt兲
for all t = 1,2, . . . ,12
共5兲
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007 / 195
where St⫽reservoir storage at the beginning of period t
共106 m3兲; It⫽inflow to the reservoir during period t 共106 m3兲;
EVPt⫽evaporation losses during period t 共106 m3兲 共a nonlinear function of initial and final storages of period t兲; and
OVFt⫽overflow from the reservoir in period t 共106 m3兲.
• Storage bounds
Smin 艋 St 艋 Smax
for all t = 1,2, . . . ,12
pQl,tHl,t 艋 El,max
for all t = 1,2, . . . ,12
共7兲
pQr,tHr,t 艋 Er,max
for all t = 1,2, . . . ,12
共8兲
pQb,tHb,t 艋 Eb,max
for all t = 1,2, . . . ,12
共9兲
where El,max, Er,max, and Eb,max⫽maximum amount of power
共106 kW h兲 that can be produced 共turbine capacity兲 from left,
right, and bed level turbines, respectively.
• Canal capacity limits
Ql,t 艋 Cl,max
for all t = 1,2, . . . ,12
共10兲
Qr,t 艋 Cr,max
for all t = 1,2, . . . ,12
共11兲
where Cl,max and Cr,max⫽maximum canal carrying capacity
limits of the left and right bank canals, respectively
共106 m3 / month兲.
• Irrigation demands
D1min,t 艋 Ql,t 艋 D1max,t
for all t = 1,2, . . . ,12
共12兲
D2min,t 艋 Qr,t 艋 D2max,t
for all t = 1,2, . . . ,12
共13兲
where D1min,t and D1max,t⫽minimum and maximum irrigation
demands for left bank canal, respectively; and D2min,t and
D2max,t⫽minimum and maximum irrigation demands for right
bank canal, respectively, in time period t.
• Water quality requirements
Qb,t 艌 MDTt
for all t = 1,2, . . . ,12
共14兲
where MDTt⫽minimum release to meet downstream water
quality requirement 共106 m3兲.
• Steady state storage constraint
S13 = S1
t=1
12
+ k2
t=1
共6兲
where Smin and Smax⫽minimum and maximum storage limits
of the reservoir.
• Maximum power production limits
共15兲
This constraint is required to bring the steady state condition
for the reservoir storage, i.e., storage at the end of a year is
equal to the initial storage at the beginning of that year.
The reservoir is mainly meant for irrigation, and so priority is
given to irrigation. After meeting the irrigation demands, power
production is to be maximized. To handle these multiple objectives, in this study a weighted approach is adopted. By giving a
larger weight to irrigation and a smaller weight to hydropower
generation, it is possible to satisfy the prescribed priorities. In this
model a normalized squared deficit is used for irrigation 共instead
of simpler squared deficits兲 with the aim to evenly distribute the
deficits throughout the season in the case of occurrence of deficits, by giving equal priority to larger and smaller magnitudes of
demands of right bank and left bank canals, respectively. To bring
both the objectives into same units, the hydropower objective is
nondimensionalized. So the final fitness function for the model is
as follows:
兺 冋冉
兺冋
12
F = k1
+
Dl,t − Ql,t
Dl,t
冊 冉
2
+
Dr,t − Qr,t
Dr,t
冊册
2
El,max − p共Ql,tHl,t兲 Er,max − p共Qr,tHr,t兲
+
El,max
Er,max
Eb,max − p共Qb,tHb,t兲
Eb,max
册
共16兲
where k1 and k2⫽constant weights to be chosen based on priority.
So the final model is to minimize F 关Eq. 共16兲兴 duly satisfying
the constraints on reservoir continuity, storage bounds, turbine
capacity, canal capacity, irrigation demands, mandatory releases,
and steady state storage constraint, i.e., Eqs. 共5兲–共15兲.
Results
The EMPSO technique is applied to the model described in the
earlier section to obtain the operating policies for the multipurpose reservoir system. In this study to handle the constraints of
the problem, simulation and evaluation approach is used, in which
at each generation, the decision variables are evaluated for the
present solution and then the bounds are checked. If there is any
violation in satisfying the constraints, then a penalty is applied by
choosing a suitable penalty coefficient. The constant weights to be
given for irrigation and hydropower can be decided by the policy
maker. In this study, since the irrigation release is having higher
priority over hydropower production, for demonstration purpose
the constant weights of the fitness function are chosen as k1⫽100
and k2 = −1. Before applying the EMPSO technique, the parameters required should be decided. The total number of decision
variables of the model is 36 共number of time periods⫽12 and
number of decisions in each period⫽3兲, which is equal to
the dimension of the problem. The termination criterion is,
whether it reaches the maximum number of iterations or if there is
no significant improvement in the solution 共up to a convergence
limit of 0.0001兲 in 50 consecutive iterations.
The sensitivity analysis of the PSO model is performed with
different combinations of each parameter. In this analysis, it is
observed that by considering a proper value for constriction coefficient 共␹兲, the inertial weight does not have much influence on
the final result of the model. So in this case study the inertial
weight 共␻兲 is fixed as 1. Also it is found that the value of constriction coefficient equal to 0.9 is yielding better results for the
given model. After a number of trials, it is found that cognitive
parameter c1 = 1.0 and social parameter c2 = 0.5 resulted in better
quality solutions. To find the optimal parameters for population
size and maximum number of iterations, a thorough sensitivity
analysis is carried out for different combinations of the parameter
settings. Fig. 4共a兲 shows the sensitivity for population size, from
which it can be observed that the minimum fitness value is at
population size of 200. Fig. 4共b兲 shows the sensitivity for maximum number of iterations and the minimum fitness value is observed for maximum iterations of 500. Similarly to determine the
number of particles on which the elitist-mutation strategy is to be
performed and the optimal number of iterations after which the
elitist-mutation strategy should be started, a thorough sensitivity
analysis is carried out. From Fig. 4共c兲, it can be observed that the
number of elitist-mutated particles is equal to 20, which leads to
better performance of the model. From Fig. 4共d兲 it can be observed that for up to 100 iterations there is no significant improve-
196 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007
Fig. 4. Sensitivity analysis of EMPSO parameters showing the relationship between: 共a兲 size of the swarm and mean fitness; 共b兲 maximum
number of iterations and mean fitness; 共c兲 influence of number of elitist-mutated particles on the mean fitness value of the model; and 共d兲 influence
of elitist-mutation 共EM兲 starting iteration on the performance of mean fitness value 共mean fitness is the average fitness value of 10 random trials兲
ment in the fitness values with the elitist-mutation step, so the
elitist-mutation step started iteration is taken as 100. Probability
of elitist mutation 共Pem兲 is adopted as 0.2. So these parameters are
adopted to run the model. The only difference between EMPSO
and the standard PSO is that the elitist-mutation strategy is performed in EMPSO. All other parameters adopted are the same for
both models.
The same reservoir operation problem is also solved using the
real coded GA technique and the results are compared with those
by the EMPSO model. After thorough sensitivity analysis, GA
model parameters selected are as follows: size of the population,
N⫽200; maximum number of generations⫽500; crossover probability, Pc = 0.85; mutation probability, Pm = 0.05; the distribution
index for simulated binary crossover⫽10 and that for mutation
operator⫽100.
To demonstrate the efficiency of the EMPSO model, the model
is first applied to a typical year, where the inflow is well represented for the reservoir catchment. Using the values of 200 for
population size and 500 for maximum number of iterations, all
the three algorithms, namely EMPSO, standard PSO, and GA,
were run for 10 different trials. Fig. 5 shows the details of
the improvement of fitness values over the iterations, which also
details the convergence properties of the EMPSO and PSO techniques. Fig. 6 shows the comparison of fitness values and number
of function evaluations required to reach the fitness values for
EMPSO, PSO, and GA models. Using the EMPSO model the best
fitness value obtained is 72.272, with average fitness value of
72.504 and standard deviation of 0.14; whereas in standard PSO,
the best fitness value is 72.972, with 80.26 and 3.85 as mean and
standard deviations, respectively, and in GA the best is 72.72,
with 73.389 and 0.479 as mean and standard deviations, respectively. For 10 trial runs, the average number of function evaluations taken to get the best solutions is 85,440, 83,280, and 95,660
for EMPSO, standard PSO, and GA, respectively. The GA model
is taking more function evaluation numbers compared to other
algorithms, and the solution quality is inferior to that of EMPSO.
It can be observed that the EMPSO is consistently giving best
fitness values over different trials than standard PSO. From these
results it is clear that EMPSO is outperforming the standard PSO
and GA models with better quality optimal solutions obtained
using less number of function evaluations.
To evaluate the performance of the proposed EMPSO in detail,
Fig. 5. Improvement of fitness value over the iterations shows the
convergence properties for EMPSO and standard PSO
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007 / 197
Fig. 6. Comparison of fitness value 共nondimensional兲 and the
number of function evolutions taken for EMPSO, standard PSO and
GA models for optimal reservoir operation problem for a typical year
of 10 random trials
the same model is applied for a good number of years and the
results are reported in this section. In the following discussions, it
may be noted that comparison is made with only the best solution
obtained in ten trials with the respective algorithm. The total annual irrigation deficits 共squared deficits兲 obtained by the releases
made to the left and right bank canals over a period of 15 years
are obtained as 141,601.94, 147,300.87, and 145,180.93 for the
EMPSO, standard PSO, and GA models, respectively. It can be
observed that EMPSO is performing better than the standard PSO
and GA techniques. Fig. 7 shows the time series plot of monthly
irrigation release made for different techniques over a period
of 15 years. The other objective of the model is to maximize
the hydropower production, for that the average annual hydropower production obtained over a period of 15 years is 1,820.71,
1,802.57, and 1 , 805.91⫻ 106 kW h by the EMPSO, PSO, and
GA techniques, respectively. Here also it can be noticed that highest hydropower production is obtained with the EMPSO technique. Fig. 8 shows the time series plot of monthly hydropower
production obtained using the EMPSO, PSO, and GA techniques.
The time series plot of monthly initial reservoir storage obtained
with the EMPSO, standard PSO, and GA techniques over a plan-
ning horizon of 15 years is shown in Fig. 9. The causes for superior performance of EMPSO over the other methods in terms of
less irrigation deficits and more hydropower production can be
summarized as apart from river bed turbine, the water released for
irrigation to the left bank canal and right bank canal also generates hydropower. If the water is released in an optimal manner, it
yields less irrigation deficits, and also contributes to hydropower.
If more water is released through the river bed, it may generate
more hydropower, but causes higher irrigation deficits. In objective function higher weightage is given for irrigation, so it will
first try to minimize the irrigation deficits, and then come to maximization of hydropower. A best optimal solution should satisfy
both these terms. So EMPSO is performing both the tasks well
and hence is resulting in lesser irrigation deficits and higher hydropower production. If any overflows occur during a time period, this also affects the final results depending on the model
performance. In the overall perspective, it is found that the operating policies obtained by the EMPSO technique are better than
those by the standard PSO and GA techniques.
Discussion
In EMPSO methodology the additional parameters involved
require little extra effort over that of standard PSO. But the
improvement in solution quality is significant with EMPSO. This
is achieved by maintaining diversity in the population and effectively exploring the search space throughout the iterations by
applying the elitist-mutation step. In EMPSO the number of particles to be mutated depends on the size of the swarm, which
again depends on the complexity of the problem. From experimental test results, it is observed that 3冑3 N 共where N⫽size of the
swarm兲 gives reasonably good results. So it can be set as a default
parameter chosen for optimization. Similarly, for elitist-mutation
starting iteration, as a default it can be performed for all the
iterations. But from experimental test results, it is noticed that
in early phases of iterations, this strategy does not have much
influence. So after a preselected number of iterations 共e.g., after
5–10% of maximum number of iterations兲, this strategy can be
introduced to effectively save computational time and to improve
efficiency of the algorithm. The probability of elitist mutation
Fig. 7. Time series plot of monthly irrigation release obtained with EMPSO, standard PSO and GA techniques over a period of 15 years 共from
1984–1985 to 1998–1999兲
198 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007
Fig. 8. Time series plot of monthly hydropower generated with EMPSO, standard PSO and GA techniques over a period of 15 years 共from
1984–1985 to 1998–1999兲
共Pem兲 ranges from 0.01 to 0.5 共depending on the complexity of
the decision space兲.
The optimal solution found by the EMPSO technique is promising. So application of the EMPSO technique for derivation of
reservoir operating policies is very much useful for real life problems, since this technique has the following advantages over other
optimization techniques:
1. The method does not depend on the nature of the function it
maximizes or minimizes. Thus, approximations made in conventional techniques are avoided.
2. EMPSO uses objective function information to guide
the search in the problem space. Therefore EMPSO can
easily deal with nondifferentiable and nonconvex objective
functions.
3. Unlike the traditional methods, the technique is not affected
by the initial searching points thus ensuring a quality solution
with high probability of obtaining the global optimum for
any initial solution.
4.
5.
6.
7.
In EMPSO, particle movement uses randomness in its
search. Hence, it is a kind of stochastic optimization algorithm that can search a complicated and uncertain area. This
makes EMPSO more flexible and robust than conventional
methods.
The convergence is not affected by the inclusion of more
constraints.
Unlike GA, EMPSO has the flexibility to control the balance
between the global and local exploration of the search space.
This property enhances the search capabilities of the PSO
technique and yields better quality solutions with fewer function evaluations.
Unlike standard PSO, EMPSO is more reliable in giving better quality solutions with reasonable computational time,
since the elitist-mutation strategy avoids the premature convergence of the search process to local optima and provides
better exploration of the search process.
Fig. 9. Time series plot of monthly initial reservoir storage obtained for EMPSO, standard PSO and GA techniques over a period of 15 years
共from 1984–1985 to 1998–1999兲
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2007 / 199
Conclusions
In this study, a new swarm intelligence algorithm, namely
EMPSO, is propounded to derive operating policies for a multipurpose reservoir system. The technique has global and local exploration capabilities to search for the optimal solutions. In order
to overcome the problems of trapping at local optima and to improve the consistency of the standard PSO algorithm in reaching
the global optimum, an elitist-mutation mechanism is incorporated into the algorithm. The proposed approach is first tested for
a hypothetical multireservoir system. The results obtained by
EMPSO compared well with results reported by DPSA, DDDP,
and FDP. When compared with standard PSO and GA techniques,
it is observed that EMPSO is consistently giving better quality
solutions, using less number of function evaluations. After
achieving satisfactory performance for the hypothetical problem,
EMPSO is applied to a realistic problem, the Bhadra reservoir
system in India to demonstrate the performance of the proposed
technique. First it is applied for a typical year and the efficiency
of the EMPSO technique is proved by comparing the obtained
results with other algorithms. Then the same model is applied
over a longer period of 15 years. It is found that the results obtained by EMPSO are better than those by the standard PSO and
GA techniques in terms of minimum irrigation deficits and maximum hydropower production. The main advantages of EMPSO
are, easy to implement, requires less number of function evaluations, and yet is efficient in obtaining global optima. The results
of this study thus amply demonstrate that the EMPSO technique
can be effectively used to solve real life optimization problems
with better efficiency.
Notation
The following symbols are used in this paper:
Cl,max , Cr,max ⫽ maximum carrying capacity of left and right
bank canals;
c1 , c2 ⫽ positive constants;
D ⫽ dimension of the solution vector;
Dl,t , Dr,t ⫽ irrigation demands of left and right bank
canal command areas, respectively, during time
period t;
E ⫽ total energy production;
El,t , Er,t , Eb,t ⫽ hydropower produced at left bank canal, right
bank canal, and bed turbines, respectively,
during time period t;
El,max , Er,max , Eb,max
⫽ maximum hydropower that can be produced
at left bank canal, right bank canal, and bed
turbines, respectively;
EVPt ⫽ evaporation losses in period t;
Hl,t , Hr,t , Hb,t ⫽ net head available at left bank canal, right
bank canal, and bed turbines, respectively,
during time period t;
It ⫽ inflow to the reservoir during time period t;
k1 , k2 ⫽ constant weight coefficients;
MDTt ⫽ mandatory release to be made in period t;
N ⫽ size of the swarm;
OVFt ⫽ overflow in period t;
Pc ⫽ probability of crossover;
Pem ⫽ probability of elitist mutation;
Pg ⫽ position of the best particle in the swarm;
Pi ⫽ best position vector of the ith particle;
Pm
p
pid
Ql,t , Qr,t , Qb,t
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
mutation probability;
power production coefficient;
best position of the ith particle dth variable;
releases made to left bank canal, right bank
canal, and bed turbines, respectively, during
time period t;
uniformly distributed random numbers;
initial storage of reservoir in time period t;
minimum and maximum storage limits of
reservoir;
sum of squared deviation;
velocity vector of the ith particle;
velocity change of the ith particle dth
variable;
solution vector of the ith particle;
decision value of the ith particle dth variable;
constriction coefficient; and
inertial weight.
⫽
⫽
⫽
⫽
index
index
index
index
r1 , r2 ⫽
St ⫽
Smin , Smax ⫽
SQDV ⫽
Vi ⫽
vid ⫽
Xi
xid
␹
␻
Subscripts
d
i
g
t
of
of
of
of
decision variable;
swarm population;
the best particle in the swarm; and
time period.
Superscripts
n ⫽ index of iteration number.
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